cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A054428 Inverse permutation to A054427.

Original entry on oeis.org

1, 2, 4, 3, 6, 9, 8, 5, 10, 17, 21, 16, 15, 20, 14, 7, 12, 27, 40, 31, 39, 54, 45, 26, 25, 44, 51, 36, 30, 35, 24, 11, 18, 41, 63, 56, 70, 101, 92, 55, 62, 114, 136, 100, 90, 113, 77, 38, 37, 76, 110, 87, 99, 133, 109, 61, 50, 85, 98, 67, 49, 60, 34, 13, 22, 57, 94, 79, 117
Offset: 1

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Author

Antti Karttunen

Keywords

Links

Programs

  • Maple
    [seq(nthmember(j, A054427),j=1..200)]; # nthmember as in A054426

A038567 Denominators in canonical bijection from positive integers to positive rationals <= 1.

Original entry on oeis.org

1, 2, 3, 3, 4, 4, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

n occurs phi(n) times (cf. A000010).
Least k such that phi(1) + phi(2) + phi(3) + ... + phi(k) >= n. - Benoit Cloitre, Sep 17 2002
Sum of numerator and denominator of fractions arranged by Cantor's ordering (1/1, 2/1, 1/2, 1/3, 3/1, 4/1, 3/2, 2/3, 1/4, 1/5, 5/1, 6/1, ...) with equivalent fractions removed. - Ron R. King, Mar 07 2009 [This applies to a(1, 2, ...) without initial term a(0) = 1 which could correspond to 0/1. - Editor's Note.]
Care has to be taken in considering the offset which may be 0 or 1 in related sequences (see crossrefs), e.g., A038568 & A038569 also have offset 0, in A038566 offset has been changed to 1. - M. F. Hasler, Oct 18 2021

Examples

			Arrange fractions by increasing denominator then by increasing numerator: 1/1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, ...: this is A038566/A038567.

References

  • S. Cook, Problem 511: An Enumeration Problem, Journal of Recreational Mathematics, Vol. 9:2 (1976-77), 137. Solution by the Problem Editor, JRM, Vol. 10:2 (1977-78), 122-123.
  • Hans Lauwerier, Fractals, Princeton University Press, 1991, p. 23.

Links

Crossrefs

Cf. A020652, A020653, A038566 - A038569, A093602, A182972, A182973 - A182976.
A054427 gives mapping to Stern-Brocot tree.
Cf. A037162.

Programs

  • Haskell
    import Data.List (genericTake)
a038567 n = a038567_list !! n
a038567_list = concatMap (\x -> genericTake (a000010 x) $ repeat x) [1..]
-- Reinhard Zumkeller, Dec 16 2013, Jul 29 2012
  • Maple
    with (numtheory): A038567 := proc (n) local sum, k; sum := 1: k := 2: while (sum < n) do: sum := sum + phi(k): k := k + 1: od: RETURN (k-1): end: # Ulrich Schimke (ulrschimke(AT)aol.com)
  • Mathematica
    a[n_] := (k = 0; While[ Total[ EulerPhi[ Range[k]]] <= n, k++]; k); Table[ a[n], {n, 0, 77}] (* Jean-François Alcover, Dec 08 2011, after Pari *)
Flatten[Table[Table[n,{EulerPhi[n]}],{n,20}]] (* Harvey P. Dale, Mar 12 2013 *)
  • PARI
    a(n)=if(n<0,0,s=1; while(sum(i=1,s,eulerphi(i))
  • Python
    from sympy import totient
def a(n):
    s=1
    while sum(totient(i) for i in range(1, s + 1))Indranil Ghosh, May 23 2017
  • Python
    from functools import lru_cache
@lru_cache(maxsize=None)
def A002088(n): # based on second formula in A018805
    if n == 0:
        return 0
    c, j = 0, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*((A002088(k1)<<1)-1)
        j, k1 = j2, n//j2
    return n*(n-1)-c+j>>1
def A038567(n):
    kmin, kmax = 0, 1
    while A002088(kmax) <= n:
        kmax <<= 1
    kmin = kmax>>1
    while True:
        kmid = kmax+kmin>>1
        if A002088(kmid) > n:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Jun 10 2025

Formula

From Henry Bottomley, Dec 18 2000: (Start)
a(n) = A020652(n) + A020653(n) for all n > 0, e.g., a(1) = 2 = 1 + 1 = A020652(1) + A020653(1). [Corrected and edited by M. F. Hasler, Dec 10 2021]
n = a(A015614(n)) = a(A002088(n)) - 1 = a(A002088(n-1)). (End)
a(n) = A002024(A169581(n)). - Reinhard Zumkeller, Dec 02 2009
a(A002088(n)) = n for n > 1. - Reinhard Zumkeller, Jul 29 2012
a(n) = A071912(2*n+1). - Reinhard Zumkeller, Dec 16 2013
a(n) ~ c * sqrt(n), where c = Pi/sqrt(3) = 1.813799... (A093602). - Amiram Eldar, Dec 27 2024

Extensions

More terms from Erich Friedman

A054424 Permutation of natural numbers: maps the canonical list of fractions (A020652/A020653) to whole Stern-Brocot (Farey) tree (top = 1/1 and both sides < 1 and > 1, but excluding the "fractions" 0/1 and 1/0).

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 5, 6, 15, 16, 31, 32, 9, 11, 12, 14, 63, 64, 10, 13, 127, 128, 17, 23, 24, 30, 255, 256, 19, 28, 511, 512, 33, 18, 20, 47, 48, 27, 29, 62, 1023, 1024, 22, 25, 2047, 2048, 65, 35, 39, 21, 95, 96, 26, 56, 60, 126, 4095, 4096, 34, 40, 55, 61, 8191, 8192
Offset: 1

Views

Author

Antti Karttunen

Keywords

Examples

			Whole Stern-Brocot tree: 1/1 1/2 2/1 1/3 2/3 3/2 3/1 1/4 2/5 3/5 3/4 4/3 5/3 5/2 4/1 1/5 2/7
Canonical fractions: 1/1 1/2 2/1 1/3 3/1 1/4 2/3 3/2 4/1 1/5 5/1 1/6 2/5 3/4 4/3 5/2 6/1

Links

Crossrefs

Cf. A047679, A007305, A007306, A054427, A057114. In table form: A054425. Inverse permutation: A054426.

Programs

  • Maple
    cfrac2binexp := proc(c) local i,e,n; n := 0; for i from 1 to nops(c) do e := c[i]; if(i = nops(c)) then e := e-1; fi; n := ((2^e)*n) + ((i mod 2)*((2^e)-1)); od; RETURN(n); end;
frac2position_in_whole_SB_tree := proc(r) local k,msb; if(1 = r) then RETURN(1); else if(r > 1) then k := cfrac2binexp(convert(r,confrac)); else k := ReflectBinTreePermutation(cfrac2binexp(convert(1/r,confrac))); fi; msb := floor_log_2(k); if(r > 1) then RETURN(k + (2^(msb+1))); else RETURN(k + (2^(msb+1)) - (2^msb)); fi; fi; end;
canonical_fractions_to_whole_SternBrocot_permutation := proc(u) local a,n,i; a := []; for n from 2 to u do for i from 1 to n-1 do if (1 = igcd(n,i)) then a := [op(a),frac2position_in_whole_SB_tree(i/(n-i))]; fi; od; od; RETURN(a); end; # ReflectBinTreePermutation and floor_log_2 given in A054429

Formula

canonical_fractions_to_whole_SternBrocot_permutation(30);
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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